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Solving Complex Equations: A Step-by-Step Guide
This article will guide you through the process of solving the complex equation (4-3i)x - (3-2i)y = 6-5i for the complex variables x and y.
Understanding Complex Numbers
Before we start, let's refresh our understanding of complex numbers. Complex numbers are expressed in the form a + bi, where:
- a is the real part
- b is the imaginary part
- i is the imaginary unit, where i² = -1
Solving the Equation
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Separate Real and Imaginary Components:
We'll rewrite the equation by separating the real and imaginary parts on both sides:
(4x - 3y) + (-3x + 2y)i = 6 - 5i
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Equating Real and Imaginary Parts:
For two complex numbers to be equal, their real and imaginary parts must be equal. Therefore, we get two separate equations:
- 4x - 3y = 6 (Equation 1)
- -3x + 2y = -5 (Equation 2)
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Solving the System of Equations:
Now we have a system of two linear equations with two unknowns. We can solve this system using various methods like substitution, elimination, or matrices. Here we'll use elimination:
- Multiply Equation 1 by 2: 8x - 6y = 12
- Multiply Equation 2 by 3: -9x + 6y = -15
- Add the two equations: -x = -3
- Solve for x: x = 3
- Substitute x = 3 into Equation 1: 12 - 3y = 6
- Solve for y: y = 2
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The Solution:
We have found the solution to the complex equation:
- x = 3
- y = 2
Verification
We can verify our solution by substituting the values of x and y back into the original equation:
(4-3i)(3) - (3-2i)(2) = 6-5i 12 - 9i - 6 + 4i = 6 - 5i 6 - 5i = 6 - 5i
The equation holds true, confirming our solution.
Conclusion
By separating real and imaginary components, we successfully solved the complex equation. This process can be applied to solve any complex equation involving multiple complex variables.